Field Inside the Harbor

In this section, we present the method to determine the field inside the harbor of large characteristic dimensions. The geometry of the problem is shown in Figure 4.1. We assume that the size of the harbor A is much larger than the width of the

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In Section 4.4, we used the scattering matrix approach to determine the field inside the harbor with dimensions comparable with the width of the entrance channel.

It was shown that if the size of the harbor becomes relatively big compared to the width of the channel, the evaluation of scattering matrix elements by the methods of Section 4.4 tends to be infeasible. To evaluate the elements of the scattering matrix, given by Eq. (4.13), we will approximate the field inside the aperture of the stepped waveguide by the field inside the aperture of the flanged waveguide. Once the scattering matrix has been evaluated, the linear system (4.11) can be solved and the field inside the harbor can then be determined.

For transmission problem on the stepped waveguide region, the field in the narrow waveguide is given as

and the field in the wider waveguide is represented as

where γn and tm are unknown coefficients to be used in scattering matrix F. The incident wave urn' is the mode coming from the narrow waveguide, zit" = ^ok,x4) j.

To determine the unknown coefficients 2^,7„ and tm, we solve the corresponding transmission problem for the same incident mode in the flanged waveguide region, and obtain the reflection coefficients a,. We then approximate the field inside the aperture of the stepped waveguide by the field field inside the aperture of the flanged waveguide. Therefore, the scattered field inside the aperture is now given as u =

E

an(1)(z). Then the scattered field in the narrow waveguide is approximated by u, as x approaches

Using the orthogonality of mode functions Is„, we obtain the reflection coefficients for the stepped waveguide region as

The continuity condition of the velocity across aperture is given as

Using zero boundary condition for the velocity on the walls of the structure together with orthogonality of T^r, in the harbor, we obtain the following solution for the transmission coefficients:

For the case of reflection problem in the stepped waveguide, the field in the narrow waveguide is expressed as

and the field in the wider waveguide is given as

The incident mode is coming from the wider waveguide

field inside the aperture is approximated by the field in the aperture of the flanged waveguide for the same incident mode u =

E a

^-

,(

^1),. We apply the orthogonality of the modes and continuity condition to determine the approximation for the trans-mission and reflection coefficients as:

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From Eqs. (5.21) and (5.24) it follows that in order to evaluate the elements of

M

x

M

scattering matrix, where

M

is a number of propagating modes, we need to obtain solutions for the flanged waveguide problem. These solutions are determined by solving

N

x

N

linear system, where N is independent of the size of the harbor.

We maintain

N =

60 for all our numerical models derived in Chapters 4 and 5.

For the purpose of our numerical simulation, we send a water wave with wavelength 26m into the channel of length 124m and width 20.7m. The channel enters large 2482m wide harbor. The scaling with respect to channel width non-dimensionalizes the data to

k =

5,

1 =

6, and a = 120. For these dimensions of the harbor, we have two modes in the entrance channel and 191 mode in the harbor basin.

Figure 5.7 The magnitude of the propagating channel and harbor modes.

In Figure 5.7, we depict the magnitude of dominating modes in the channel, Ro, and in the harbor, Ao and A190. We note that the standing mode dominates the higher order modes. We also see that these zero order modes in the channel and harbor almost perfectly match each other. The remaining energy is redistributed among the other modes. Due to the computational complexity of the problem, we

limited the range of the length of the harbor to [170, 180, and located resonances in this range. We can see two resonant lengths at

b =

173.7 and

b =

179.35 in both channel and harbor regions.

Figure 5.8 shows the field inside the harbor of resonant dimensions a = 120,

b =

179.3. We note the pattern of standing waves propagating inside the harbor basin.

Due to the scattering matrix method used to obtain the field, the representation of the field is not accurate near

x =

0. Nevertheless, we can clearly see a broadening channel inside the harbor with almost no field within the channel. The waves disperse away from this shadow region.

Figure 5.8 The field inside the harbor of length

b =

^179.35.

The development of the calm region in the middle of the harbor can be explained by the following argument. The field in the harbor is excited by the modes coming through the entrance channel. The first mode can be decomposed into two plane waves that strike the opposite corners of the aperture at the same angles. The

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waves diffracted off the opposite corners are similar to each other in frequency and magnitude but differ in phase. Due to this phase shift, the superposition of these two diffracted fields can interfere and cancel each other inside the region bordered by two outgoing plane waves representing the incident mode.

Figure 5.9 shows the field inside the harbor basin for another resonant length of b = 173.7. To be able to see wave pattern more clearly, we pictured only the central part of the harbor. For this harbor width, we can also note the shadow channel in the middle of the harbor, however, the field is now present inside the channel. This field is due to the reflections from the boundaries of the harbor.

Figures 5.9 and 5.8 demonstrate that the method presented in this Chapter produces physically plausible results for the field inside the harbor of large dimensions.

The asymptotic approximation employed in this method allowed us to cut the amount of calculations by nearly a⁴.

FUTURE RESEARCH

In document Electronic Theses and Dissertations (Page 86-92)